All Questions
Tagged with teaching linear-algebra
17 questions
22
votes
2
answers
2k
views
Can one deduce the fundamental theorem of algebra from real calculus and linear algebra?
Motivation: let $A\in\mathbf{R}^{n\times n}$ be symmetric. Then by the method of Lagrange multipliers, a maximum of $x\mapsto x^tAx$ on the compact unit sphere $\mathbf{S}^{n-1}$ must be an ...
- 337
5
votes
1
answer
521
views
How to find eigenvalues following Axler?
Preparing my Linear Algebra lecture I like the determinant free approach of Axler because the proof that operators $T$ on an $n$-dimensional complex vector space have eigenvalues is so simple:
Fix ...
- 16.5k
0
votes
1
answer
114
views
Name of a matrix with one column and row removed [closed]
I am looking for the exact name of a matrix where the i-th column and rows have been removed.
I cannot remember how it is called in linear algebra, does anyone got an idea?
Thanks!
- 111
35
votes
2
answers
2k
views
Is it consistent with ZF that $V \to V^{\ast \ast}$ is always an isomorphism?
Let $k$ be a field and $V$ a $k$-vector space. Then there is a map $V \to V^{\ast \ast}$, where $V^{\ast}$ is the dual vector space. If we are in ZFC and $\dim V$ is infinite, then this map is not ...
- 156k
49
votes
14
answers
21k
views
Applications of the Cayley-Hamilton theorem
The Cayley-Hamilton theorem is usually presented in standard undergraduate courses in linear algebra as an important result. Recall that it says that any square matrix is a "root" of its own ...
34
votes
6
answers
3k
views
Does seeing beyond the course you teach matter? The case of linear algebra and matrices
This question is indeed very important for me. Thus I hope you bear with my subjective explanations for a few minutes. I am an "excellent" lecturer, at least according to course evaluation forms ...
2
votes
0
answers
1k
views
Linear Algebra Text Book [closed]
In our department we do not like our current linear algebra book and so we would want to find a better book. This is for the first course in linear algebra and the title of the course is
Elementary ...
2
votes
4
answers
1k
views
Eigenvalues of powers of linear mappings
Let $\tau$ be a linear map on a finite dimensional complex vector space. Clearly, if $\lambda$ is an eigenvalue of $\tau$ then $\lambda^n$ is an eigenvalue of $\tau^n$, for any natural (integer, on ...
- 4,096
19
votes
3
answers
2k
views
Research level applications of "row rank = column rank"?
No less an authority than Gilbert Strang frames "row rank equals column rank" (and a couple of other facts) as "The Fundamental Theorem of Linear Algebra."
I'd simply like to assemble (for teaching ...
34
votes
13
answers
6k
views
Elementary applications of linear algebra over finite fields
I'm teaching axiomatic linear algebra again this semester. Although the textbooks I'm using do everything over the real or complex numbers, for various reasons I prefer to work over an arbitrary ...
5
votes
3
answers
2k
views
Continuous change of basis (and on the definition of determinant) [closed]
Let $(u_1, \ldots, u_n)$ and $(v_1, \ldots, v_n)$ be two ordered bases of $\mathbb R^n$. The orientation of the first basis is defined as the sign of the determinant of $[u_1 \cdots u_n]$, and ...
- 1,174
5
votes
1
answer
1k
views
Is Diagonalization worth to be taught? [closed]
When students come to the College (first two years of the University system in most of the developped countries) to train in mathematics, they get a linear algebra / matrix analysis course. After a ...
5
votes
5
answers
2k
views
Topics for a matrix analysis course
I recently taught a new (to my department) course titled "Matrix Analysis". For various reasons that I won't go into here, I was dissatisfied with the textbook I (loosely) followed, and with every ...
14
votes
2
answers
7k
views
What is the dual concept to "annihilator" called, and do any linear algebra textbooks discuss this concept first?
When introducing dual spaces for the first time, most linear algebra textbooks proceed in what seems to me a rather backwards fashion: the annihilator $\{f\in V^*: f(u)=0\quad \forall u\in U\}$ of a ...
10
votes
7
answers
2k
views
Proof that bases etc. exist in early linear algebra course?
I'm currently struggling to teach a 2nd course on linear algebra (in the UK, not at an Oxbridge quality university: the students have done a 1st course which concentrated upon algorithms you can apply ...
51
votes
22
answers
19k
views
Why linear algebra is fun!(or ?)
Edit: the original poster is Menny, but the question is CW; the first-person pronoun refers to Menny, not to the most recent editor.
I'm doing an introductory talk on linear algebra with the ...
23
votes
13
answers
7k
views
Pedagogical question about linear algebra
Last semester I taught a linear algebra class that is intended to introduce young students (at a sophmore-junior level) to "abstract mathematics". It seems that a major conceptual hurdle for many of ...